Support Vector Data Description by using hyper-ellipse instead of hyper-sphere

Abstract

Abstract —Support Vector Data Description (SVDD) describes data by using a hyper-sphere. In this paper, we propose an extended SVDD (ESVDD) which describes data by using a hyper-ellipse. Clearly, ESVDD can describe data better than SVDD in the input space. Both hyper-sphere and hyper-ellipse are very rigid for data description. The kernel ESVDD which will be proposed in this paper and the kernel SVDD enhance the ability of ESVDD and SVDD for data description, respectively. The formulation of SVDD/ESVDD contains a penalty term C which controls the tradeoff between the volume of hyper-sphere/hyper-ellipse and the training errors. We show that the ESVDD can control this tradeoff better than the SVDD. Keywords- Kernel; ESVDD; Data description; Hyper-ellipse. I. I NTRODUCTION The one-class classification problem is an interesting field in pattern recognition and machine learning researches. In this kind of classification, we assume the one class of data as the target class and the rest of data are classified as the outlier. One-class classification is particularly significant in applications where only a single class of data objects is applicable and easy to obtain. Objects from the other classes could be too difficult or expensive to be made available. So we would only describe the target class to separate it from the outlier class. The SVDD is a kind of one-class classification method based on Support Vector Machine [1, 2] which proposed by Tax. It tries to construct a boundary around the target data by enclosing the target data within a minimum hyper-sphere. Inspired by the support vector machines (SVMs), the SVDD decision boundary is described by a few target objects, known as support vectors (SVs). A more flexible boundary can be obtained with the introduction of kernel functions, by which data are mapped into a high-dimensional feature space. The most commonly used kernel function is Gaussian kernel. This method has attracted many researchers from the various fields. For example Liu et al. applied the SVDD techniques for novelty detection as part of the validation on an Intelligent Flight Control System (IFCS) [3]. Ji et al. discussed the SVDD application in gene expression data clustering [4]. Yu et al. used SVDD for image categorization from internet images [5]. Recently, some efforts have been expended to improve the SVDD method. Guo et al. proposed a simple post-processing method which tries to modify the SVDD boundary in order to achieve a tight data description [6]. As another example Cho apply the orthogonal filtering as a preprocessing step is executed before SVDD modeling to remove the unwanted variation of data [7]. In this paper, we propose an extended SVDD (ESVDD) which describes data by using a hyper-ellipse instead of a hyper-sphere. Clearly, ESVDD can